Modern Stochastics: Theory and Applications

Let ${({\xi _{k}},{\eta _{k}})_{k\ge 1}}$ be independent copies of a random vector $(\xi ,\eta )$ with positive arbitrarily dependent components. Put and then The random sequences $S:={({S_{k}})_{k\ge 0}}$ and $T:={({T_{k}})_{k\ge 1}}$ are known in the literature as the standard random walk and a (globally) perturbed random walk. A survey of various results for the so defined perturbed random walks can be found in the book [9].

Put A law of the iterated logarithm (LIL) for $Y(t)$, properly normalized and centered, was proved as $t\to \infty $ along integers in Proposition 2.3 of [11] under the assumptions that $\mathbb{E}{\eta ^{a}}\lt \infty $ for some $a\gt 0$ and ${\sigma ^{2}}:=\mathrm{Var}\hspace{0.1667em}\xi \in (0,\infty )$. We improve the aforementioned result by showing that the assumption $\mathbb{E}{\eta ^{a}}\lt \infty $ for some $a\gt 0$ can be dispensed with and also that the LIL holds as $t\to \infty $ along reals, thereby obtaining an ultimate version of the LIL for $Y(t)$. For a family $({x_{t}})$ of real numbers denote by $C(({x_{t}}))$ the set of its limit points.

Next, we consider a general branching process generated by the random sequence ${({T_{k}})_{k\ge 1}}$. Thus, the random variables ${T_{1}},{T_{2}},\dots $ are interpreted as the birth times of the first generation individuals. The first generation produces the second generation. The shifts of birth times of the second generation individuals with respect to their mothers’ birth times are distributed according to copies of T, and for different mothers these copies are independent. The second generation produces the third one, and so on.

Let ${Y_{j}}(t)$ be the number of the jth generation individuals with birth times $\le t$. Following [3], we call the sequence of processes ${({({Y_{j}}(t))_{t\ge 0}})_{j\ge 2}}$ an iterated perturbed random walk. Note that, for $t\ge 0$, ${Y_{1}}(t)=Y(t)$ and the following decomposition holds: where ${Y_{j-1}^{(r)}}(t)$ is the number of the jth generation individuals who are descendants of the first generation individual with birth time ${T_{r}}$. Put $V(t):={V_{1}}(t)=\mathbb{E}Y(t)$ for $t\ge 0$. Taking expectations in (1) we infer, for $j\ge 2$ and $t\ge 0$,

The iterated perturbed random walks are interesting objects on their own, see [14, 16]. Also, these are the main auxiliary tool in investigations of nested infinite occupancy schemes in random environment. Details can be found in the papers [4–6, 15]. Attention was also paid to iterated standard random walks, which are a rather particular instance of the iterated perturbed random walks which corresponds to $\eta =\xi $. An LIL for the iterated standard random walks was recently proved in [12]. Continuing this line of investigation we formulate and prove an LIL for ${Y_{j}}(t)$, properly normalized and centered, as $t\to \infty $.

Although the beginning of our proof of Theorem 2 is similar to that of Theorem 1.1 in [12], the subsequent technical details are essentially different. The main difficulty is that the distribution of η is arbitrary. Imposing a moment assumption on the distribution of η would greatly simplify an argument.

The remainder of the paper is structured as follows. After proving Theorem 1 in Section 2, we give a number of auxiliary results in Section 3 and then prove Theorem 2 in Section 4.

We shall denote by n an integer argument and by t a real argument. For $t\in \mathbb{R}$, put $F(t):=\mathbb{P}\{\eta \le t\}$ and and observe that $F(t)=0$ and $\nu (t)=0$ for $t\lt 0$. For $t\gt \mathrm{e}$, write and put $a(t):={(2{\sigma ^{2}}{\mu ^{-3}}t\log \log t)^{1/2}}$. It is shown in the proof of Proposition 2.3 in [11] that This result holds irrespective of whether $\mathbb{E}{\eta ^{a}}\lt \infty $ for some $a\gt 0$ or $\mathbb{E}{\eta ^{a}}=\infty $ for all $a\gt 0$. We intend to show that (5) entails Given $t\ge 4$ there exists $n\in \mathbb{N}$ such that $t\in (n-1,n]$. Hence, by monotonicity, Analogously, We conclude that (6) does indeed hold.

It is known (see the proof of Theorem 3.2 in [1]) that whenever $\mathbb{E}{\eta ^{a}}\lt \infty $ for some $a\gt 0$. We note that the latter limit relation may fail to hold if $\mathbb{E}{\eta ^{a}}=\infty $ for all $a\gt 0$. For instance, it follows from Remark 4.4 in [13] that the upper limit in the last displayed formula is equal to $+\infty $ a.s. whenever $\mathbb{P}\{\xi =c\}=1$ for some $c\gt 0$ and ${\lim \nolimits_{t\to \infty }}(\log \log t)(1-F(t))=1$.

The proof of Theorem 3.2 in [1] operates with power moments and relies heavily upon the assumption $\mathbb{E}{\eta ^{a}}\lt \infty $ for some $a\gt 0$. Without such an assumption another argument is needed, which operates with exponential rather than power moments. In the remainder of the proof we present such an argument, which enables us to prove that thereby completing the proof of the theorem; here, $b(t):={(t\log \log t)^{1/2}}$ for $t\gt \mathrm{e}$.

Fix any $u\ne 0$ and $t\gt 0$. Put ${W_{0}}:=1$ and, for $j\in \mathbb{N}$, and denote by ${\mathcal{G}_{0}}$ the trivial σ-algebra and, for $j\in \mathbb{N}$, by ${\mathcal{G}_{j}}$ the σ-algebra generated by ${({\xi _{k}},{\eta _{k}})_{1\le k\le j}}$. Observe that the variable ${W_{j}}$ is ${\mathcal{G}_{j}}$-measurable for $j\in {\mathbb{N}_{0}}:=\mathbb{N}\cup \{0\}$. Now we prove that ${({W_{j}},{\mathcal{G}_{j}})_{j\ge 0}}$ is a positive supermartingale. Indeed, writing ${\mathbb{E}_{j}}(\cdot )$ for $\mathbb{E}(\cdot |{\mathcal{G}_{j}})$ and using the inequality ${\mathrm{e}^{x}}\le 1+x+{x^{2}}{\mathrm{e}^{|x|}}/2$ for $x\in \mathbb{R}$ in combination with we infer In view of $|{1_{\{{T_{j}}\le t\}}}-F(t-{S_{j-1}}){1_{\{{S_{j-1}}\le t\}}}|\le 1$ a.s., the right-hand side does not exceed For the latter inequality we have used $1+x\le {\mathrm{e}^{x}}$ for $x\ge 0$. Thus, we have proved that, for $j\in \mathbb{N}$, ${\mathbb{E}_{j-1}}({W_{j}}/{W_{j-1}})\le 1$ a.s. and thereupon ${\mathbb{E}_{j-1}}{W_{j}}\le {W_{j-1}}$ a.s., that is, ${({W_{j}},{\mathcal{G}_{j}})_{j\ge 0}}$ is indeed a positive supermartingale. As a consequence, the a.s. limit satisfies $\mathbb{E}{W_{\infty }}\le \mathbb{E}{W_{0}}=1$. In other words, with $u\in \mathbb{R}$ and $t\gt 0$ fixed,

We shall also need another auxiliary result.

Fix any $\varepsilon \gt 0$ and put ${t_{n}}:=\exp ({n^{3/4}})$ for $n\in \mathbb{N}$. We intend to prove that To this end, for $n\ge 3$, define the event In view of (9), for large n, Using this we obtain, for any $u\gt 0$ and large n, Invoking Markov’s inequality in combination with (8) we infer Let $\rho \gt 0$ satisfy $\exp (8{\varepsilon ^{-1}}\rho )=3/2$. For large $x\gt 0$, ${x^{-1}}\log \log x\le \rho $. Put Then Hence, by the Borel–Cantelli lemma, ${\limsup _{n\to \infty }}(X({t_{n}})/b({t_{n}}))\le 0$ a.s. The converse inequality for the lower limit follows analogously. We start with ${A_{n}^{\ast }}:=\{-X({t_{n}})\gt \varepsilon b({t_{n}})\}$ and show, by the same reasoning as above, that ${A_{n}^{\ast }}\subseteq {B_{n}^{\ast }}$, where ${B_{n}^{\ast }}$ only differs from ${B_{n}}$ by the term $-uX({t_{n}})$ in place of $uX({t_{n}})$.

It remains to show that (10) can be lifted to (7). To this end, it suffices to prove that Indeed, (11) in combination with (10) entails This ensures (7) because, for large enough n, whenever $t\in [{t_{n}},\hspace{0.1667em}{t_{n+1}}]$.

We denote by $I={I_{n}}$ a sequence of positive integers to be chosen later. For $j\in {\mathbb{N}_{0}}$ and $n\in \mathbb{N}$, put In what follows, we write ${v_{j,\hspace{0.1667em}m}}$ for ${v_{j,\hspace{0.1667em}m}}(n)$. Observe that ${F_{j}}(n)\subseteq {F_{j+1}}(n)$. For any $u\in [{t_{n}},{t_{n+1}}]$, put An important observation is that either ${u_{j-1}}={u_{j}}$ or ${u_{j-1}}={u_{j}}-{2^{-j}}({t_{n+1}}-{t_{n}})$. Necessarily, ${u_{j}}={v_{j,m}}$ for some $0\le m\le {2^{j}}$, so that either ${u_{j-1}}={v_{j,m}}$ or ${u_{j-1}}={v_{j,m-1}}$. Write For $u\in {F_{I}}(n)$, With this at hand, we obtain

We first show that, for all $\varepsilon \gt 0$, Let $\ell \in \mathbb{N}$. As a preparation, we derive an appropriate upper bound for $\mathbb{E}{(X(u)-X(v))^{2\ell }}$ for $u,v\gt 0$, $u\gt v$. Observe that $X(u)-X(v)$ is equal to the a.s. limit ${\lim \nolimits_{j\to \infty }}R(j,u,v)$, where ${(R(j,u,v),{\mathcal{G}_{j}})_{j\ge 0}}$ is a martingale defined by and, as before, ${\mathcal{G}_{0}}$ denotes the trivial σ-algebra and, for $j\in \mathbb{N}$, ${\mathcal{G}_{j}}$ denotes the σ-algebra generated by ${({\xi _{k}},{\eta _{k}})_{1\le k\le j}}$. Recall that $F(t)=0$ for $t\lt 0$. By the Burkholder–Davis–Gundy inequality, see, for instance, Theorem 11.3.2 in [7], for a positive constant C. Let $f:[0,\infty )\to [0,\infty )$ be a locally bounded function. It is shown in the proof of Lemma A.3 in [1] that $\mathbb{E}{(\nu (1))^{\ell }}\lt \infty $ and that Further, Using (14) with $t=u$ and $f(y)={1_{[0,\hspace{0.1667em}u-v)}}(y)$ and then with $t=v$ and $f(y)=F(u-v+y)-F(y)$ we infer where $x\mapsto \lceil x\rceil $ is the ceiling function, and Finally, and thereupon Note that ${v_{j,\hspace{0.1667em}m}}-{v_{j,\hspace{0.1667em}m-1}}={2^{-j}}({t_{n+1}}-{t_{n}})$. Put $I={I_{n}}:=\lfloor {\log _{2}}({2^{-1}}({t_{n+1}}-{t_{n}}))\rfloor $. We claim that there exists a constant ${C_{2}}\gt 0$ such that ${C_{1}}{(\lceil {2^{-j}}({t_{n+1}}-{t_{n}})\rceil +1)^{\ell }}\le {C_{2}}{2^{-j\ell }}{({t_{n+1}}-{t_{n}})^{\ell }}$ whenever $j\in \mathbb{N}$, $j\le I$. Indeed, having utilized ${2^{-j}}({t_{n+1}}-{t_{n}})\ge 2$ for $j\le I$. Invoking (15) we then obtain, for nonnegative integer $j\le I$, and thereupon By the triangle inequality for the ${L_{2\ell }}$-norm, By Markov’s inequality, Since ${t_{n}^{-1}}({t_{n+1}}-{t_{n}})\sim (3/4){n^{-1/4}}$ as $n\to \infty $, (13) follows upon setting $\ell =6$, say. Invoking the Borel–Cantelli lemma we infer

Now we proceed with the analysis of the second summand in (12). Put $M(t):={\textstyle\int _{[0,\hspace{0.1667em}t]}}F(t-y)\mathrm{d}\nu (y)$ for $t\ge 0$. Using the equality $X(t)=N(t)-M(t)$ and a.s. monotonicity of N and M we infer Observe that Hence, according to the Borel–Cantelli lemma, it is enough to prove that, for all $\varepsilon \gt 0$, and Arguing as above we conclude that, for $u,v\gt 0$, $u\gt v$, As a consequence, for nonnegative integer $j\le I$ and a constant ${C_{4}}\gt 0$, By Markov’s inequality and our choice of I, Hence, (17) follows upon choosing $\ell \gt 2$. To prove (18), we invoke (16) which enables us to conclude that Choosing $\ell \gt 1$ we arrive at (18).

The proof of Theorem 1 is complete.

To prove Theorem 2, we need some auxiliary results on the iterated perturbed random walks. Lemma 1 is a known result, see Assertion 1 in [16].

Put so that U is the renewal function.

We shall also need two results on the standard random walks. The next lemma is a consequence of formula (33) in [1], with $\eta =\xi $.

We use a decomposition The first term of the decomposition is treated in Proposition 1.

We first prove Theorem 2 with the help of Proposition 1. Afterwards, a proof of Proposition 1 will be given.

Finally, we prove Proposition 1.